Some archery physics

by Bas ten Hoope

(last change: October 8th, 1995)

The author does not claim to be complete, he's just trying to provide a starting point for mathing archery.

He uses knowledge gathered during years of shooting, talking about archery, discussing innovations, and doing some private modelling. All of which follow his own interests and may therefore stop

somewhere unexpected quite abruptly...


Launching an arrow

Arrow flight

Stabilising equipment

Launching an arrow

General considerations

Today's common recurved bow consists of a rigid handle in which two flexible limbs are placed. The limbs are forced towards the archer by means of a string, yielding the more commonly seen picture of a strung bow.

What gives the recurved bow its name is the shape of the limbs. Instead of being straight pieces of wood when unstrung, the tips of the limbs are curved away from the archer, they are 'recurved'. When the bow is strung, there is still ample recurve left in the limbs. For an undrawn bow, in practice this produces the sight of the string being laid over the limbs near their ends. When the bow is drawn, this contact is soon ended, and gradually more and more of the initial recurved shape is drawn out of the limbs. The shape seems to have developed through history and today the recurved bow clearly is superior in single string target archery.

Today bows are made in all relevant sizes and drawing weights. The technique in target archery requires the archer to decide on how heavy a limb he can draw, while still being able to focus and aim at the target. Most archers choose the heaviest limbs (mind we're talking about drawing weight) they can still draw with enough ease to allow them to produce a correct release.

As said earlier, when the bow is drawn further more and more of the recurved shape is drawn out of the limbs. For the bow to remain as efficient as possible, a certain degree of recurve must remain in the limb. Every target archer having a personal value for his/her maximum drawing length and maximum drawing weight, he/she buys those limbs that yield the desired weight at their personal drawing length. The ideal limb size is thus related to the archers' body.

After this it should be clear that the recurved bows superiority over other bows nowadays in target archery, does not come from how heavy the limbs can be made, but from the high efficiency with which the archers labour is transferred to the arrow. The efficiency of a bow can be said to be due to the limbs shape combined with the archers shooting technique.

To make a tiny side-step: The concept of the so called 'compound bow', is a mechanical trick which allows the archer to have a far larger energy locked up in the limbs, as compared to recurved or normal bows, while he/she can still comfortably aim before releasing.

The influence of the archers technique on bow efficiency, is not a direct influence as such. I state it here to keep the reader from getting to much at ease with the dynamic picture of an ideal release. The bow is drawn by fingers hooked around the string, thus it is impossible (for human beings at least) to release the string without giving it a slight displacement in the horizontal direction (the bow is held vertical). In fact, this always occurring initial displacement is seen to make the arrow buckle in the right direction while it is accelerated by the string. To be brief: Keep in mind that the bow design must be dynamically insensitive to common mistakes of the ever imperfect archer. As it is unlikely that being able to draw a bow was a factor in the evolution of mankind, our bodies are simply not ideally suited for it.

By far the largest part of the bows efficiency however is buried in its shape and general flexibility, or where exactly it bends how much. Nowadays non-rigid handles and strings are thought to be very inconvenient, and usually they are made as stiff as possible. For the handle different alloys containing ever lighter elements are used to make a firm handle. Concessions towards rigidity to get the handles' mass lower may become common (though currently not done), as bow mass is a major factor in the archers' convenience during the shooting. Today's strings are made of what chemistry can invent that is strong, not elastic and very light. As for dynamic reasons the string mass must be kept as low as possible. Every part of the bow that still has speed while the arrow is no longer accelerated cuts in the bows efficiency.

Today's performance limbs are made from carbon or whatever stuff that is very endurable, can be made very hard to bend (so less material may be used to gain the same drawing weight!), and has a low hysteresis or energy consumption due to friction or whatever. In general the ratio: mass / energy-it-can-hold should be as low as possible for the material the limb is made from.

The major problems in efficiency thus are energy loss in the limbs due to friction and redundant movement of limb mass as the arrow leaves the bow. The energy of this limb movement clearly being no longer helpful to the arrow.

Since both together yield the efficiency one could even decide to take a material with a higher internal friction which is so much lighter, that the over all efficiency still improves.

Now to incorporate limb shape in the discussion of bow efficiency. The limb grows thinner towards its end, which is about where the recurve is as well. The thin recurved tip is what does the major movement of the limb as the bow is drawn, the broad part near the handle bends only a small way. What results is that a relative small mass does most movement, Whatever kinetic energy the limb possesses after the arrow departs from the string is lost. The recurve shaped limb does a great job in keeping those losses minimal, by keeping the mass of what is moving fast minimal.

Further reading: 'On the mechanics of the bow and arrow' B.W. Kooi 1983

A simple model for the recurved bow

Measurement of the static Drawing Force - Drawing Length diagram of a recurved bow yields an almost straight line within the Drawing Length area the bow is meant for.

So the model of a recurved bow can be made real simple and to a good approximation by a straight line from (0,0) towards (personal Drawing length, personal Drawing weight). The area below this line is Force times Distance integrated or the Work the archer has done on his bow. Ideally, all of this should enter the arrow, but we already expected it not to. For an approximation of the resulting arrow velocity at launch, I'll just use an efficiency percentage, based on overall bow quality or - if you wish - on speculation.

So, for my bow this would imply:

personal Drawing length: L = 0.52 m

Drawing weight at that position:

  P = 40 # = 40 * .45 * g = 180 N 

Arrow mass: m = 0.021 kg

over all Efficiency: e = .70

With arrow velocity v this becomes:

  * m * v^2 =  * e * P * L 

This would give a Launch velocity of 56 m/s ( 200 km/h ; 180 ft/s ), a number I will use later. Now how did I get this 0.7 efficiency ratio ? I've got carbon limbs and assumed my major losses to be due to mass accelerated not being the arrow. My strings mass is about 0.0075 kg so I added just a bit more limb mass to get a round figure of 0.030 kg effectively accelerated mass. For a crude approximation I'll still wager it to be within 10% of its actual value. This because the initial velocity leads to a trajectory later on, which seems to be in agreement with observed trajectories.

Mind though, I'm primarily interested in a bit of modelling. I never tried to measure actual arrow speeds, or actual arrow trajectories. But hey, trust me, it should be about right. Uhh...

Another thing that can be calculated this way is how long it actually ought to take the arrow to get launched. This is an important number if one for instance wishes to assert whether a certain stabiliser of a certain known stiffness, can be expected to do anything at all. I myself was simply curious.

To do this, use an effective mass for the arrow and with the modelled linear force-draw function:

  F = q * (L-x)  
( F force , q constant , x starting position )

dividing by mass :

  a = F/m = (q/m) * (L-x) 
( a acceleration )

So, we got the second order time derivative of the place ( x ) to be equal to a constant times something like the place. This is a second order inhomogeneous differential equation. Its homogeneous solution is easily seen to be:

 x = c1 * cos (sqrt(q/m) * t) + c2 * sin (sqrt(q/m) * t) 

t being elapsed time after releasing. A particular solution would be: x = L , so the general solution becomes:

 x = c1 * cos (sqrt(q/m) * t) + c2 * sin (sqrt(q/m) * t) + L

Using now that at departure the string is at x=0 and its velocity v=0 as well:

 At t=0 : x(0) = c1 + L      = 0     , so:  c1 = -L
 At t=0 : v(0) = sqrt(q/m) * c2 = 0     , so:  c2 = 0

This finally yields our displacement function (place, velocity, acceleration) :

 x = L * ( 1 - cos (sqrt(q/m) * t) )
 v = L * sqrt(q/m) * sin (sqrt(q/m) * t)
 a = L * q/m * cos (sqrt(q/m) * t)

So, if on x(T)=L the arrow leaves the arrow,

 cos (sqrt(q/m) * T) = 0 
which means that
  sqrt(q/m) * T =  p 
  T =  p sqrt(m/q) 
. In my case, since q has to be
  q = P / L = 180/0.52 
, this gives us T = 0.012 s .

To check if we arrive at the same velocity using T in the velocity function gives us

  v(T) = L * sqrt(q/m) = sqrt( P * L / m ) 
which is exactly the same formula as derived earlier for the launch velocity if we were to replace the arrow mass by
  m / e 
, the effective mass.

So everything fits and we get an ideal launch time of about 12 milliseconds. In Listons book however (see reference far down), it is mentioned that actual releasing takes a bit longer as the fingers mess with the string. I guess what matters here, is where one defines the string to be released or free to go. From there it'll be some 12 milliseconds.
During the relaxing of the fingers, the bow pulls the string free as the drawing arm pulls the hand backwards. The releasing process is a major source of error in target archery.
If one were to optimise stabilising equipment, one might want to keep the bow as stable as possible for the total of releasing time plus arrow acceleration time. The next problem would be to estimate releasing time, but not here...

The compound bow

After having discussed the drawing behaviour of the recurved bow, an interesting extension can be made by looking at the so called compound bow.

A compound bow is a bow that uses an acentrically placed wheel at the top of both thick limbs to drastically increase the energy storage in the limbs, compared to what can be stored in the limbs of a single string bow.

As said before, the energy the archer can store in the limbs is limited by the force that can still successfully be handled by the archer when the string is fully drawn. The same still holds for the compound bow user, it's the draw-force relationship that accounts for the difference in energy storage.

Typical compound bows have a so called 'let-off' of about 50 percent. Which means that about halfway between undrawn and fully drawn one has to apply twice the force needed to hold the bow in fully drawn position. The purpose is evident, compared to the recurved bow one can store more than twice as much energy in the limbs.

graph of Static Force-draw diagram

Now, how is this Force-Draw trajectory obtained? Both limbs are straight and very sturdy, only about two thirds the length of a recurved limb, and stuck in the handle at about 20 degrees with the vertical handle, facing the archer.

(picture of a compound bow, invented by Wilbur Allen)

At the tip of the limbs is a small incision in which a wheel is placed, connected to the limb by means of an axis. Both wheels have been manipulated to constitute two strings, and are each others mirror image.

The first string is guided over the wheel's edge around the wheel and is the one the archer will use to draw the bow. It is fastened on the bottom part of the top wheel, then led backwards over the wheel over the top before stretching down towards the archers touch. From there it's the same backwards: down, around the wheel and fastened at the top of the bottom wheel.

Now to grasp what's going on imagine the wheels to resist rotation by a constant torque (moment of force) per unit of rotation. As a model, we imagine this 'rotation-spring' to be the only energy container of the bow and thus take both limbs rigid. For the time being, forget I mentioned a second string.

Effectively, we hold one wheel at its rotation axis, and accelerate a projectile as if we hold a catapult, drawing horizontally towards us.

If the wheel is rotated by alpha degrees, the torque would be a constant times alpha. If the wheel would be rotating about its center, the arm would be constant and equal to the wheel radius. (Start a line where the string leaves the wheel, draw the line in the direction of the applied force (F). The arm will be the shortest distance of this line towards the wheel's rotation axis.)

If we were to pull at the string now, we would get:

F * arm = F * radius  =  torque  =  constant * alpha 

which means :

F  =  constant * alpha / radius  =  constant2 * alpha

that is:

F increases linearly with alpha.

Thus the Draw-Force curve would be a straight line, the static behaviour of this model of the compound bow would be the same as the static recurved bow's model.

Now to show what the compound bow's about, we displace the rotation axis from the center. We move the wheel horizontally, a distance q towards the archer (Who will, for the time being, still be taken to be some brat with a catapult-like device).

Now, this time the arm varies with the drawing length, as the string is drawn. Initially, at alpha = 0, the arm is equal to the wheel radius: [ r ].

Drawing, the arm shortens to [ r - q ] at alpha = 90 degrees.

From there, it increases again to [ r ] at 180 degrees,

and continues to increase to [ r + q ] at alpha = 270 degrees.

For our (compound) purpose we need draw no further, what happened during drawing is the following. At all times:

F * arm  =  constant * alpha

However, we've seen the arm to be a function of alpha. To gain an expression of F as a function of alpha, we need the arm as a function of alpha. So we go through some math: (I'll use a 2-dimensional plane: x horizontal, y vertical. The string is drawn horizontally (in the +x direction))

The wheel, a circle centered at (x,y)=(a,b) :

(x-a) squared + (y-b) squared = r squared 
        ( => sqr(x-a)+sqr(y-b)= sqr(r) )

a and b are dislocated from 0 as the circle rotates about (-q,0) by an angle alpha ( in the formulas: t = alpha ) :

sqr( x - q(cos[t]-1) ) + sqr( y - q(sin[t]) ) = sqr( r )

Now, for each t we need to know the largest y value possible, as this will be the topmost part of the wheel, and where the string leaves the wheel horizontally: (sqrt = square root)

         arm  =  maximum(y)
for y > 0 : y = - q(sin[t]) + sqrt{sqr(r)- sqr(x - q(cos[t]-1))}

The maximum of y is achieved at x = q(cos[t]-1) :

  maximum(y)  =  r - q(sin[t])

Or: the arm = r - q sin[ alpha ]

Which yields our force:

F = c * alpha/(r - q sin[alpha])

When we plot this force as a function of alpha, with parameter q at r/2 or more, and extend it from alpha = 0 to alpha = 270 degrees, then we get about the shape as shown in the graph. The compound bow curve.

That is: there is a maximum for the applied force somewhere near alpha = 90 degrees! Actually we need to know the force as a function of the drawing length to know the energy. This would mean that the graph of F(alpha) would be stretched somewhere and condensed elsewhere to become F(drawing length).

But that's just a slight transformation of the alpha-axis, the shape will remain the same: a curve with a maximum at about halfway drawn.

We now see what would have happened if we would have drawn further. Again we would have entered the 'short arm' region, and the force would have risen more than linearly, and probably above our might.

This concludes the model, in reality the wheels hold no torque towards their axis. The limbs take care of the necessary 'torque', that is: by bending they produce the force to counter the archer's applied force. For this to occur however, we need something more, as will be shown imminently:

If we took the compound bow as far as we built it before the catapult model started: {limb + acentriccally placed wheels + one string fastened at the wheels}, then pulling at the string would result in the effortless rotation of both wheels, and no limb movement. When we keep drawing, the wheels unwind, and suddenly when they're unwound we have still no energy stored yet, but are already directly pulling at the stiff limbs. That's not what we want !

To make sure there's some kind of energy storage like in the model, a second string is used. The effect of this string should be to force the limbs to bend when the wheels turn as the archer pulls. The string does this by shortening the distance between the top and bottom wheel.

The string can be taken as a single wire, (In reality it is not, but this way the effect is more easily seen.) which is fastened at the same (mirrored) positions on both wheels: At the top wheel, the string is fixed below the rotation point. The bottom wheel connects to the string at the same point but above its rotation axis. Now, both connecting points move up resp. down as both wheels are rotated towards the archer, thus forcing both limbs to bend towards each other and store energy.

In reality the second string is a double string and I'm not going to pretend I'm sure why. I only watched a compound bow to grasp the construction of its peculiar Draw-Force curve. There are more minor details, which I will not mention, as they are not vital to the bow's drawing behaviour. Mostly, I suppose, they are adjustments and additions to make the whole construction a bit less sensitive to releasing- or other mistakes by the archer. I might update this article with more information on the actual lay-out of a compound bow, but as I'm no compound archer it might take a whole lot of a time for me to stumble across the info...

In general, one can see that forcing the second string to bend around material, added for that purpose on the wheel, manipulates the shape of drawing-force curve, as it influences how much the limbs will have to bend in what area of drawing length. Another factor is the shape of the wheel. It's already acentrically placed, why keep it a circle? What I mean to show is that there must be an optimum for the entire drawing- and shooting behaviour of the compound bow. There are so many parameters, but I've said enough for now.

(know no more to say, did not think about it anymore to say anything, do not wish to think about it anymore or say anything about it at all, etc...)

At least I should have accomplished explaining how the neat Force-Draw behaviour of the compound bow is achieved, if not in overly much detail.

I'll finish with an inaccurate (I'm not a compound archer, after all) description of an archer using a compound bow.

The archer swiftly pulls through the heavy draw section, then settles into the easy draw section and takes his/her time to fine tune aiming before releasing. Whack!

(Frustrated remark of a recurve user, hearing all the violence and seeing grouping he can only dream of :)

But even though I'm not a compound archer, I can still see the deception supported by the bow. The large accumulated energies ready to suffuse the arrow lures many archers to neglect what's of paramount importance in archery: a proper technique during the entire process that makes the arrows fly.